Abstract
We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained.
Similar content being viewed by others
References
Aussel D., Hadjisavvas N.: On quasimonotone variational inequalities. J. Optim. Theory Appl. 121, 445–450 (2004)
Bianchi M., Pini R.: Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124, 79–92 (2005)
Bianchi M., Hadjisavvas N., Schaible S.: Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities. J. Optim. Theory Appl. 122, 1–17 (2004)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Brézis H., Nirenberg L., Stampacchia G.: A Remark on Ky Fan’s minimax principle. Bolletino della Unione Matematica Italiana 6, 293–300 (1972)
Browder F.E.: Nonlinear Monotone operators and convex sets in Banach space. Bull. Am. Math. Soc. 71, 780–785 (1965)
Chipot M.: Inequalities and Flow in Porus Media, Appl. Math. Sci. 52. Springer-Verlag, New York (1984)
Chowdhury M.S.R., Tan K.K.: Generalization of Ky Fan’s minimax inequality with applications to generalized variational inequalities for pseudomonotone operators and fixed-point theorem. J. Math. Anal. Appl. 204, 910–929 (1996)
Cottle R.W., Giannessi F., Lions J.L. (eds): Variational Inequalities and Complementarity Problems—Theory and Applications. Wiley, New York (1980)
Daniilidis A., Hadjisavvas N.: On generalized cyclically monotone operators and proper quasimonotonicity. Optimization 47, 123–135 (2000)
Ding X.P., Tarafdar E.: Generalized variational-like inequalities with pseudomonotone set-valued mappings. Archiv der Mathematik 74, 302–313 (2000)
Fakhar M., Zafarani J.: Generalized equilibrium problems for quasimonotone and pseudomonotone bifunctions. J. Optim. Theory Appl. 123, 349–364 (2004)
Fakhar M., Zafarani J.: Generalized vector equilibrium problems for pseudomonotone multi-valued bifunctions. J. Optim. Theory Appl. 126, 109–124 (2005)
Fan K.: A Generalization of Tychonoff’s fixed-point theorem. Math. Ann. 142, 305–310 (1961)
Farajzadeh, A., Zafarani, J.: Equilibrium problems and variational inequalities in topological vector spaces (to appear in Optimization)
Hadjisavvas N.: Continuity and maximality properties of pseudomonotone operators. J. Convex Anal. 10, 465–475 (2003)
Hadjisavvas N., Schaible S.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90, 95–111 (1996)
Isac G.: Complementarity Problems, Lecture Notes in Mathematics 1528. Springer-Verlag, New York (1991)
Kalmoun E.M.: On Ky Fan’s minimax inequalities, mixed equilibrium problems and hemivariational inequalities. J. Inequal. Pure Appl. Math. 2, 1–13 (2001)
Karamadian S., Schaible S.: Seven kinds of monotone maps. J. Optim. Theory Appl. 66, 37–46 (1990)
Kien B.T., Yao J.-C., Yen N.D.: On the solution existence of pseudomonotone variational inequalities. J. Glob. Optim. 41, 135–145 (2008)
Kinderleher D., Stampacchia G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York (1980)
Yao J.C., Guo J.S.: Variational and generalized variational inequalities with discontinuous mappings. J. Math. Anal. Appl. 182, 371–392 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of T. Rapcsák.
The first author was partially supported by a grant from IPM (No. 86470016).
Rights and permissions
About this article
Cite this article
Fakhar, M., Zafarani, J. On generalized variational inequalities. J Glob Optim 43, 503–511 (2009). https://doi.org/10.1007/s10898-008-9346-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-008-9346-2